Correlation and Causation

Review - 1• Two types of correlational study

– When same items have values on two score variables, correlate the scores on one with the scores on the other

• Measure degree of correlation in terms of Pearson coefficient r

• Predict value on one variable from that on the other using the regression line: y=ax+b

– When one nominal variable divides a population into two or more sub-populations, compare the two (or more) populations on another (score) variable in terms of their central tendencies

• If the means are different, predict the value on the score variable depending on the value of the nominal variable

Review - 2• In both types of correlational studies, one commonly

makes inferences from a sample to an actual (total) population

– Does what is found in the sample apply to the actual population?

– Addressed in terms of statistical significance • Is the result in the sample one that would be

unlikely to happen by chance if there weren’t a correlation or a difference in the actual population?

• The p value specifies the likelihood of the result in the sample happening by chance (in drawing the sample)

–p < .05 indicates there is less than 5% chance of the result happening by chance

Clicker QuestionA study based on a sample of 100 UCSD students reported a difference in interest in partying between men and women (p<.01)

A. This result is not reliable because of the small sample size

B. This result is not reliable because of the small p-value C. There is less than 1 in 100 likelihood that there is a

difference in the actual population D. There is less than 1 in 100 likelihood that the difference in

the sample is due to chance

Review - 3• In testing a claim about differences in the means of two sub-

populations, one tests the null-hypothesis – There is no difference in the means

• The strategy is to try to reject the null hypothesis using the results in the sample – If the difference in means in the sample is statistically

significant (at a chosen level), one infers that the null hypothesis is false

• Therefore, the means differ in the real populations – If the differences in means in the sample are not

statistically significant (at the chosen level), one cannot reject the null hypothesis

• Whatever differences there might be, they will not have been detected.

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Clicker QuestionIf the attempt to find a difference in means based on a sample is reported to be non-significant, that means

A.The probability that the null hypothesis was true was greater than 5%

B.The probability that the null hypothesis was true was less than 5%

C.There is no difference between the means in the actual population

D.The result is not important

Review -3 *!*

• No significant difference does not mean there is no difference

– There may well be a difference, but one that has not been detected given the tests employed

– All we can say is that we have not detected any difference

• Compare (better, contrast) – We have not found the person who killed the Prime

Minister – No one killed the Prime Minister

Caught Between Two Errors• Type I error (over confidence): Thinking there is a

difference between means when there is none – Use higher significance levels: instead of requiring

only p<.05, require p<.01 or even p<.001

• Type II error (humility): Thinking there is no difference between means when there is one – Use a larger sample, which has a greater chance of

finding a significant difference if one is to be found

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Two dangers - 2

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Ho is true Ho is false

Did not reject Ho

Correct failure to

reject

Type II error (β)

Did reject Ho Type I error (α)

Correct rejection

α and β levels• α-level is the probability of rejecting the null hypothesis

when it is true – Statistical significance and p-value

• β-level is the probability of failing to reject the null hypothesis when it is false – (1- β) is probability that the researcher will correctly

reject the null when the null is indeed false – The statistical power of the test

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Two types of error - 2

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Reduce type I error by increasing p- value

Increase Type II error

Increase sample size to reduce Type II error

Type II errorType I error

Difference in means between samples

Populations that could have produced mean in sample

Clicker QuestionUnder what conditions should one focus more on reducing type II errors than type I errors?

A. When it is critical not to claim a difference when there isn’t one

B. One should always be more concerned with type I errors

C. When it is critical not to miss a difference when there is one

D. When there is little worry about being wrong

Clicker QuestionIn which type of situation would you most likely expect that a Type II error has been committed?

A. When the difference between means in a small or moderate-sized sample is not found to be statistically significant

B. When an extremely large sample has been used C. When the difference between means in a sample has

been found to be significant (p<.01) D. When the difference between means in an extremely

large sample is not found to be statistically significant

Clicker QuestionTo reduce the likelihood of a Type II error, one should

1. Always insist on using p-values <.01 2. Not worry about the p-value and just look at the

differences produced in the sample 3. Use a large enough sample so that if there is a

difference, it will produce a significant difference in the sample

4. Use a small sample since then if there is a significant difference, there is likely to be a large difference in the real population

Science without Error?• One can reduce the risk of type I and type II errors to

whatever level one desires – If one is willing to use a large enough sample

• But one cannot eliminate the risk of error – It is always possible that there is no difference in

means despite obtaining a significant result in one’s sample

– It is always possible that there is a real difference in means, but the difference in the sample is not significant

• This is one more example of how scientific knowledge remains fallible!

Clicker QuestionIs the following a good argument for confirming a correlational claim based on a sample:

If there is a difference between means in the population, the result in the sample will be statistically significant (p<.X)

The result in the sample is statistically significant (p<.X) ∴There is a difference between means in the population A. Yes, the argument is valid B. Yes, the argument is sound C. No, the argument affirms the consequent D. No, the argument denies the antecedent

The Logic of Correlational Research

• To confirm or falsify a correlational claim based on a sample, we use modus tollens. The first premise in each case, though, is different

• Confirming a correlational claim: If there is no difference between means in the

population, then there will not be a statistically significant (p < ?) difference in my sample

There is a statistically significant difference (p < ?) in means in my sample

∴There is a difference between means in the population • We pick the level of significance in the first premise

according to how great a risk of error in our conclusion we can accept

The Logic of Correlational Research - 2

• Falsifying a correlational claim If there is a detectable difference between means in

the population, then there will be a statistically significant difference (p < ?) in my sample

There is no statistically significant difference (p < ?) in means in my sample

∴There is no detectable difference between means in the population

• The truth of the first premise depends upon using a large enough sample

• NOTE: The conclusion refers to DETECTABLE differences

Quest for finding causes• When something happens, we ask “Why?” We want to know

what caused the event – Why are we interested in causes?

• Knowing the causes frequently provides understanding

• Knowing causes empowers us to intervene • These two tend to go together

–Why do these barrels produce better beer? » Learning the reason is more hops provides

understanding » And a procedure for making better beer

–How does HIV cause AIDS? » Knowing about protease inhibitors explains » And tells us a good place to intervene

What is a cause?

• The roots of talk of causation is found in our doing something to produce an effect – We want to move a rock, so we push it – We want to see a friend so we walk to her apartment – We want to stay warm so we put on a jacket

• Independent of our own action, a cause is something which brings about or increases the likelihood of an effect – The cause of the explosion was the spark from the

generator

Correlation and Causation

• A major reason people are interested in correlations is that they might be indicative of causation

• Correlations per se only allow you to predict – The correlation of unprotected sex with having a

baby nine months later allows someone who has unprotected sex to predict that they are more likely to have a baby nine months later

• Causation tells you how to change the effect – Knowing that unprotected sex causes (increases

the likelihood of) having a baby nine months later allows you to take action to have or not have a baby

Correlations Point to Causation• Statistical relations between variables that exceed

what is statistically expected are typically due to causal relations – Although not necessary direct causal relations

• Examples: – Consumption of red wine and reduced heart

attacks – Books that have a green cover and books that do

not sell many copies – Good study habits and good grades

Major Difference: Correlation Symmetrical; Causation Asymmetrical

• Being run into in a traffic accident might be a cause for the big dent in your car

• Having a big dent in your car is correlated with having a car accident, but it is not the cause of having a car accident

• Causation is directional, correlation is symmetrical – So when correlation points to causation, we still

need to establish the direction

Challenge of Establishing Directionality

• Does watching violence on TV result in aggressive behavior in children?

• Or do the factors that generate aggressive behavior cause children to watch more violence on TV

Causal Loops• Sometimes X causes Y and then Y causes more X

– The causation here is still directional, but works in both directions

• Back pain may be the cause of a person limping – but walking with a limp may cause further back

pain

Snoring and Obesity• There is a positive correlation between obesity and

snoring • Does obesity cause (increased) snoring?

– Yes—via fat buildup in the back of the throat • But fat build up also causes sleep apnea

– Sleeper stops breathing momentarily and wakes up • As a result of sleep apnea, sufferer is tired and avoids

physical activity – Thereby getting more obese

Obesity

Snoring

Sleep Apnea

Relating Correlation and Causation

• Establishing correlation does not establish causation – But it is a big part of the project!

• If X causes Y, then one expects a correlation between X and Y – The greater the value of X (if X is a score variable),

the greater the value of Y – Individuals exhibiting X (if X is a nominal variable)

will have greater values of Y

Independent/Dependent Variables

• Independent variable –The variable that is thought to be the cause –The variable that is altered/manipulated in an

experiment –The treatment in a clinical trial

• Dependent variable –The variable that is thought to be the effect –The variable that one is trying to predict/explain –The outcome in a clinical trial

• The dependent variable depends on the independent variable

Clicker QuestionIf average driving speed is the independent variable in an

experiment then A. Its value depends upon the dependent variable B. It is the variable that is manipulated in the

experiment C. It is the variable that is affected by the manipulation D. It is to be explained by finding the cause

Measured versus Manipulated• The strongest tests of causation claims involve

manipulation of variables à Experiments • In some contexts, a researcher does not or cannot

manipulate the independent variable – Immoral to assign people to categories such as

having unprotected sex – Cannot assign people to categories such as being

female • If we are nonetheless considering causes in such a

case, we refer to a measured independent variable • When it is possible to manipulate the independent

variable (conduct an experiment), we speak of a manipulated independent variable

Clicker QuestionWhich of the following makes no sense?

A. Manipulated independent variable B. Measured independent variable C. Manipulated dependent variable D. Measured dependent variable

Measures (Operational Definitions) and Data

• Often causal relations are specified in general terms: – Violence on TV causes violent behavior in school

• The variables used to operationally define such variables are sometimes referred to as measures. The specific values on these variables are data – “The number of gun firings on a given TV show is a

good measure of violence on the show. We have related data on gun firings to data on two measures of aggressive behavior by those watching the show.”

• The measure: Violence operationally defined as # of gun firings

• Data on # of gun firings

Correlation without direct causation

• Sometimes one variable is directly related causally to another

• But sometimes the causation is via some other link

Correlations without direct causation

• Ice cream sales and the number of shark attacks on swimmers are correlated

• SAT scores and college grades are correlated

• Skirt lengths and stock prices are highly correlated (as stock prices go up, skirt lengths get shorter).

• The number of cavities in elementary school children and vocabulary size have a strong positive correlation

When causation suspected

• Driving red cars is positively correlated with having traffic accidents

• Why? Several possible causal scenarios – accident-prone drivers prefer red – people become more aggressive when driving red

cars – more dangerous cars tend to be painted red (sports

cars) – the color red is harder to see and is more likely to be

involved in a 2-car accident – the color red is easier to see, and that leads more

drivers to steer towards the red car

Country Music and Suicide

• Out of 49 metropolitan areas studied, suicide rates are significantly higher in those in which more country music is played on the radio – Does listening to country music cause suicides? – Or?

• Suicidal people choose to live in cities with more country music played on the radio

• Country music is popular in cities with high poverty levels and it is the latter that causes higher suicide rates

• Or?

Extraneous Variables• Given the number of possible variables to consider, in

any given inquiry some variables will be correlated with the dependent variable of interest

• If these are not the variables we are focusing on, we term them extraneous

• But – What we term extraneous may in fact be the

causally relevant variable – So, in testing a causal hypothesis, care must be

taken to rule out any causal link between these extraneous variables and the dependent variable

Limits of correlation• Fluoride in water is correlated with

lower rate of tooth decay • But why?

– Fluoride reduces cavities – People in cities with fluoride enjoy better diets – People in cities with fluoride practice better dental

hygiene – People in cities with fluoride have better genetics – Water in cities with fluoride contains other minerals

(calcium) that help prevent tooth decay • These additional variables are extraneous from the point

of view of the first hypothesis, but they might be the true causes

Telling Causal Stories Can be Fun

• Correlation: Amount of ice cream sold correlates with increased deaths by drowning:

“Increases in nuclear power generator accidents (Chernobyl, Three Mile Island...) have resulted in greenhouse gas increases, ozone layer reduction, average world temperature rise and increases in the fraction of heavy water in rain. Concerns about nuclear catastrophe have resulted in increases in eating disorders, especially among those with a genetic predisposition to obesity. Heavy water in rain has resulted in an increase in the specific gravity of cream produced by cows, while the increasing world temperature has resulted in an increasing attendance at beach resorts, coupled with increased consumption of ice cream. The increased weight of fat worried people whose centre of gravity has been lowered by a rising consumption of heavy ice cream has caused an increased number of deaths by drowning.” Dr. Paul Gardner, Monash University, Australia

Telling Causal Stories can be Fun - 2

• Correlation: Number of fire trucks and amount of fire damage:

“While this could be another case of intentionally starting fires in effort to attract the fire people, this seems highly unlikely. Firefighter salaries are modest. The only logical explanation is that the community just feels so darn safe knowing that there are more fire trucks around, that they simply are not as careful and concerned with fire safety. They feel so confident that a truck would rescue them in an instant, before a fire could spread very far, so they are just careless. With this inappropriate assumption and subsequent increase in fires, the firefighters are even less able to arrive at a scene on time. Thus, more damage occurs.” Katie Brandt, Purdue University Indianapolis

Beyond causal story telling• If a causal relation exists between two variables, then if

we can directly manipulate values on one (the independent variable), we should change values on the other (the dependent variable)

• An experiment is precisely an attempt to demonstrate causal relations by manipulating the independent variable and measuring the change on the dependent variable.

Clicker QuestionDoes the following argument represent the logic of experimental confirmation?

If X is a cause of Y, then there will be a statistically significant difference in Y when X is present

There is a statistically significant difference in in Y when X is present

∴X is the cause of Y A. No, the first premise is usually false B. No, one cannot determine statistical significance in an

experiment C. No, the argument affirms the consequent D. No, the argument form is modus ponens whereas modus

tollens should be used

The Logic of Causal Research• To confirm or falsify a causal claim based on a correlation,

we use modus tollens. The first premise in each case, though, is different

• Confirming a causal claim: If X is not a cause of Y [and there is no alternative

plausible hypothesis], then there will not be a statistically significant difference in Y when X is present

There is a statistically significant difference in Y when X is present [and there is no alternative plausible hypothesis]

∴X is a cause of Y • Whether the first premise is true depends critically on how

we set up the test of the causal hypothesis—whether we make it very unlikely that anything else could produce a difference in Y

The Logic of Causal Research - 2

• Falsifying a causal claim If X were the cause of Y [and auxiliary assumptions

are true and the experimental set up is adequate], then there would be a statistically significant difference in Y when X is present

There is no statistically significant difference in Y when X is present [and auxiliary assumptions are true and the experimental set up is adequate]

∴X is not the cause of Y • The truth of the first premise depends critically on how

we set up the test of the causal claim